Let HCI be the Hua construction of the first type. We describe the Einstein-Kahler metric for HCI. We reduce the Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary function X(z,w,ζ). This differential equation can be solved to give an implicit function in x(z,w,ζ). For some cases, we obtained the solution of the differential equation and the explicit forms of the complete Einstein-Kahler metrics on HCI which are the non-homogeneous domains.